#8772991

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#56413	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,7)(3,4,8,6)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26)(27,29)(28,32)(30,34)(31,36)(33,38)(35,39)(37,40);(1,3,5,8)(2,6,7,4)(10,12)(13,21)(15,17)(19,24)(25,27,30,36)(26,31,34,29)(28,35,38,40)(32,37,33,39);(1,3)(2,4)(5,8)(6,7)(10,12)(11,18)(16,20)(19,24)(25,28,30,38)(26,32,34,33)(27,35,36,40)(29,39,31,37);(1,4)(2,3)(5,6)(7,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,34)(36,40);(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,29)(26,27)(28,37)(30,31)(32,40)(33,35)(34,36)(38,39);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,27)(26,29)(28,35)(30,36)(31,34)(32,39)(33,37)(38,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8680187);